Symmetry is one of the greatest unifying themes in modern physics and plays a fundamental role in classifying quantum phase of matter.
In condensed matter physics, the existence of periodical table in topological free fermions is a great example in which the anomalous quantum Hall effect, topological insulators and topological superconductors, and many other interesting topological phenomena are unified into a single framework of classification theory. [1]}, [2]}, [3]}
Despite high degree of universality on the classification theory, the previous researches about symmetry analysis and topology in condensed matters mainly focused on the isolated systems[4]}, [5]}, [6]}, [7]}, [8]}, [9]} or those systems coupled to non-Hermitian potentials[10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}.
However, realistic quantum systems inevitably couple to external degrees of freedom, which, in general, should be described by open quantum systems.
Recently, efforts have been made to study the classification problem on the steady states in Lindblad equations[18]}, [19]}.
The key point for experimentally searching stable symmetry-protected topological states is maintaining the coherence of quantum state in presence of surroundings.
Thus, a natural question is whether these symmetry protected topological states can survive from the decoherence induced by environment.
In particular, one of the central issues for open quantum system is the decoherence dynamics concerning how the quantum coherence evolves and vanishes which is particularly important for the quantum information and quantum computation[20]}. A crucial question is how to avoid decoherence which is unavoidable roadblock to quantum information processing.
